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Can swell increase the number of freak waves in a wind sea?

Published online by Cambridge University Press:  19 March 2010

ODIN GRAMSTAD  and
KARSTEN TRULSEN
ODIN GRAMSTAD
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
KARSTEN TRULSEN*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
*
†Email address for correspondence: karstent@math.uio.no
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Abstract

The effect of a swell on the statistical distribution of a directional short-wave field is investigated. Starting from Zakharov's spectral formulation, we derive a new modified nonlinear Schrödinger equation appropriate for the nonlinear evolution of a narrow-banded spectrum of short waves influenced by a swell. The swell-modified equation is solved analytically to yield an extended version of the result of Longuet-Higgins & Stewart (J. Fluid Mech., vol. 8, no. 4, 1960, pp. 565–583) for the modulation of a short wave riding on a longer wave. Numerical Monte Carlo simulations of the long-term evolution of a spectrum of short waves in the presence of a monochromatic swell are employed to extract statistical distributions of freak waves among the short waves. We find evidence that a realistic short-crested wind sea can on average experience a small increase in freak wave probability because of a swell provided the swell is not orthogonal to the wind waves. For orthogonal swell and wind waves we find evidence that there is almost no significant change in the probability of freak waves in the wind sea. If the short waves are unrealistically long crested, such that the Benjamin–Feir index serves as indicator for freak waves (Gramstad & Trulsen, J. Fluid Mech., vol. 582, 2007, pp. 463–472), it appears that the swell has much smaller relative influence on the probability of freak waves than in the short-crested case.

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Papers
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Journal of Fluid Mechanics , Volume 650 , 10 May 2010 , pp. 57 - 79
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Akhmediev, N., Ankiewicz, A. & Taki, M. 2009 Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373 (6), 675–678.CrossRefGoogle Scholar
Alber, I. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A 363 (1715), 525–546.Google Scholar
Craik, A. D. D. 1988 Interaction of a short-wave field with a dominant long-wave in deep-water – derivation from Zakharov's spectral formulation. J. Austral. Math. Soc. B 29, 430–439.CrossRefGoogle Scholar
Dysthe, K., Krogstad, H. E. & Müller, P. 2008 Oceanic rogue waves. Annu. Rev. Fluid Mech. 40 (1), 287–310.CrossRefGoogle Scholar
Dysthe, K. B. & Trulsen, K. 1999 Note on breather type solutions of the NLS as models for freak-waves. Phys. Scripta T82, 48–52.CrossRefGoogle Scholar
Fuhrman, D. R., Madsen, P. A. & Bingham, H. B. 2006 Numerical simulation of lowest-order short-crested wave instabilities. J. Fluid Mech. 563, 415–441.CrossRefGoogle Scholar
Gramstad, O. 2006 Kan dønning framprovosere ekstreme bølger i vindsjø? (in Norwegian) Master's thesis, Mechanic Division, Department of Mathematics, University of Oslo, Oslo, Norway.Google Scholar
Gramstad, O. & Trulsen, K. 2007 Influence of crest and group length on the occurrence of freak waves. J. Fluid Mech. 582, 463–472.CrossRefGoogle Scholar
Grimshaw, R. 1988 The modulation of short gravity-waves by long waves or currents. J. Austral. Math. Soc. B 29, 410–429.CrossRefGoogle Scholar
Henyey, F. S., Creamer, D. B., Dysthe, K. B., Schult, R. L. & Wright, J. A. 1988 The energy and action of small waves riding on large waves. J. Fluid Mech. 189, 443–462.CrossRefGoogle Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33 (4), 863–884.2.0.CO;2>CrossRefGoogle Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface-waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Lechuga, A. 2006 Were freak waves involved in the sinking of the tanker ‘Prestige’? Nat. Haz. Earth Sys. 6 (6), 973–978.CrossRefGoogle Scholar
Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger-equation. J. Fluid Mech. 150, 395–416.CrossRefGoogle Scholar
Lo, E. Y. & Mei, C. C. 1987 Slow evolution of nonlinear deep-water waves in two horizontal directions – a numerical study. Wave Mot. 9 (3), 245–259.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface-waves on longer gravity-waves. J. Fluid Mech. 177, 293–306.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8 (4), 565–583.CrossRefGoogle Scholar
Masson, D. 1993 On the nonlinear coupling between swell and wind waves. J. Phys. Oceanogr. 23 (6), 1249–1258.2.0.CO;2>CrossRefGoogle Scholar
Naciri, M. & Mei, C. C. 1992 Evolution of a short surface-wave on a very long surface-wave of finite-amplitude. J. Fluid Mech. 235, 415–452.CrossRefGoogle Scholar
Naciri, M. & Mei, C. C. 1993 Evolution of short gravity-waves on long gravity-waves. Phys. Fluids A 5 (8), 1869–1878.CrossRefGoogle Scholar
Naciri, M. & Mei, C. C. 1994 Two-dimensional modulation and instability of a short wave riding on a finite-amplitude long wave. Wave Mot. 20 (3), 211–232.CrossRefGoogle Scholar
Onorato, M., Cavaleri, L., Fouques, S., Gramstad, O., Janssen, P. A. E. M., Monbaliu, J., Osborne, A. R., Pakozdi, C., Serio, M., Stansberg, C. T., Toffoli, A. & Trulsen, K. 2009 a Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin. J. Fluid Mech. 627, 235–257.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R. & Serio, M. 2002 Extreme wave events in directional, random oceanic sea states. Phys. Fluids 14 (4), L25–L28.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R. & Serio, M. 2006 Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys. Rev. Lett. 96 (1), 014503.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. & Bertone, S. 2001 Freak waves in random oceanic sea states. Phys. Rev. Lett. 86 (25), 5831–5834.CrossRefGoogle ScholarPubMed
Onorato, M., Osborne, A. R., Serio, M., Cavaleri, L., Brandini, C. & Stansberg, C. T. 2004 Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments. Phys. Rev. E 70 (6), 067302.CrossRefGoogle ScholarPubMed
Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P. A. E. M., Kinoshita, T., Monbaliu, J., Mori, N., Osborne, A. R., Serio, M., Stansberg, C. T., Tamura, H. & Trulsen, K. 2009 b Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Phys. Rev. Lett. 102 (11), 114502.CrossRefGoogle ScholarPubMed
Regev, A., Agnon, Y., Stiassnie, M. & Gramstad, O. 2008 Sea–swell interaction as a mechanism for the generation of freak waves. Phys. Fluids 20 (11), 112102.CrossRefGoogle Scholar
Shukla, P. K., Kourakis, I., Eliasson, B., Marklund, M. & Stenflo, L. 2006 Instability and evolution of nonlinearly interacting water waves. Phys. Rev. Lett. 97 (9), 094501.CrossRefGoogle ScholarPubMed
Socquet-Juglard, H., Dysthe, K., Trulsen, K., Krogstad, H. E. & Liu, J. D. 2005 Probability distributions of surface gravity waves during spectral changes. J. Fluid Mech. 542, 195–216.CrossRefGoogle Scholar
Stansberg, C. T. 1994 Effects from directionality and spectral bandwidth on nonlinear spatial modulations of deep-water surface gravity wave trains. In 24th International Conference on Coastal Engineering, Kobe, Japan.Google Scholar
Stiassnie, M. 1984 Note on the modified nonlinear Schrödinger-equation for deep-water waves. Wave Mot. 6 (4), 431–433.CrossRefGoogle Scholar
Tamura, H., Waseda, T. & Miyazawa, Y. 2009 Freakish sea state and swell–windsea coupling: numerical study of the Suwa-Maru incident. Geophys. Res. Lett. 36, L01607.CrossRefGoogle Scholar
Toffoli, A., Lefevre, J. M., Bitner-Gregersen, E. & Monbaliu, J. 2005 Towards the identification of warning criteria: analysis of a ship accident database. Appl. Ocean Res. 27 (6), 281–291.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Mot. 24 (3), 281–289.CrossRefGoogle Scholar
Waseda, T. 2006 Impact of directionality on the extreme wave occurrence in a discrete random wave system. In Ninth International Workshop on Wave Hindcasting and Forecasting, Victoria, BC, Canada.Google Scholar
Weideman, J. A. C. & Herbst, B. M. 1986 Split-step methods for the solution of the nonlinear Schrödinger-equation. SIAM J. Numer. Anal. 23, 485–507.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
Zhang, J. & Melville, W. K. 1990 Evolution of weakly nonlinear short waves riding on long gravity-waves. J. Fluid Mech. 214, 321–346.CrossRefGoogle Scholar
Zhang, J. & Melville, W. K. 1992 On the stability of weakly nonlinear short waves on finite-amplitude long gravity-waves. J. Fluid Mech. 243, 51–72.CrossRefGoogle Scholar